**Functional analysis** is the branch of mathematics where vector spaces and operators on them are in focus. In** linear algebra**, the discussion is about ﬁnite dimensional vector spaces over any ﬁeld of scalars. The functions are linear mappings which can be viewed as matrices with scalar entries. If the functions are mappings from a vector space to itself, the functions are called operators and they are represented by square matrices. In functional analysis, the vector spaces are in general inﬁnite dimensional and not all operators on them can be represented by matrices. Hence the theory becomes more complicated, but nonetheless there are many similarities.Functional analysis has its origin in ordinary and partial diﬀerential equations, and in the beginning of the 20th century it started to form a discipline of its own via integral equations…

*Contents*

1 Introduction

2 Diﬀerential equations

2.1 Linear ordinary differential equations

2.2 Partial diﬀerential equations

2.3 Spectral theory

3 **Integral equations**

3.1 Origins in applications

3.2 Potential theory and electrostatics

3.3 The connection between Differential and Integral equations

4 Passing to the inﬁnite

4.1 Pre-linear algebra

4.2 Inﬁnite systems and determinants

4.3 General theory

5 The concept of space in the 19th century

6 Fredholm on Integral equations

7 Hilbert on Spectral theory

8 Finalizing the concept of space

8.1 A new way of mathematics – Abstraction prior to problem solving

8.2 Adding structure to abstract spaces – The introduction of Topology

9 Fr´echet on metric spaces

9.1 Synthetic geometry – Euclidean geometry in function spaces

10 Lebesgue on Integration theory

11 The creation of modern Functional Analysis

11.1 Spectral theory of compact operators

A Solution of the Dirichlet and Neumann problems by Fredholm’s meto

A.1 The interior Dirichlet problem

A.2 The exterior Dirichlet problem

A.3 The interior Neumann problem

A.4 The exterior Neumann problem

Author: Lindström, Jens

Source: Uppsala University Library

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